Quite a bit, actually. Columbus has enough genuinely free attractions to fill a long weekend without spending anything on admission.
The Columbus Museum of Art on Broad Street offers free admission every Sunday. The Wexner Center for the Arts on the Ohio State University campus charges no admission for its public galleries. The Short North Arts District along High Street north of downtown is free to walk, browse, and explore any day of the week, with galleries that typically don't charge entry. German Village, a 233-acre historic neighborhood south of downtown, costs nothing to walk through, and Schiller Park at its center is open year-round at no charge.
For outdoor space, Scioto Audubon Metro Park near the Scioto Mile has a free 62-foot climbing wall that's open to the public. The Columbus Topiary Park in the Old Towne East neighborhood, which recreates Georges Seurat's A Sunday on La Grande Jatte in sculpted greenery, is free and open daily.
The Ohio Statehouse on Capitol Square offers free self-guided tours of the building and grounds. Goodale Park in Italian Village and the Olentangy Trail along the river are also free to use year-round.
Parking costs vary by neighborhood, so the Short North and downtown areas are easiest to reach free via COTA bus routes running along High Street.